Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al. and Baekler et al., emerges also in the framework of Diakonov et al. (2011). We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann-Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) Only in a Riemann-Cartan spacetime, that is, in a spacetime with torsion, parity violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann-Cartan spacetime is a natural habitat for chiral fermionic matter fields.PACS numbers: 04.50. Kd, 11.30.Er, 98.80.Jk ‡ We follow the conventions of [6]. We have the coframe 1-form ϑ α = e i α dx i and the frame vectors e β = e j β ∂ j , with e β ⌋ϑ α = δ α β . Greek indices are raised and lowered by means of the Minkowski metric g αβ = diag (−1, 1, 1, 1). The volume 4-form is denoted by η, and η α = ⋆ ϑ α , η αβ = ⋆ ϑ αβ , η αβγ = ⋆ ϑ αβγ , η αβγδ = ⋆ ϑ αβγδ , where ⋆ is the Hodge star operator and ϑ αβ := ϑ α ∧ ϑ β , etc. § The second minus sign on the right-hand-side of this equation is corrected. In [6], Eq.(5.9.18) was a sign error. * By using some simple algebra, Eq.(16) can alternatively be derived in a straightforward way from Eqs.(10.17) to (10.22) of Obukhov [4]. and curvature invariants with even and odd parity including all boundary terms